Abstract

A full study of parity-broken states in the directional solidification of lamellar eutectics is performed within the boundary-integral formulation. Symmetric states cease to exist at a wavelength \ensuremath{\lambda}, which is approximately twice that corresponding to their minimum undercooling, whereas solutions with a broken parity, drifting transversely to the growth front, appear as a forward bifurcation. Our results suggest that if one effectively doubles the wavelength of the initially symmetric state---a situation that can be achieved via a sudden jump of the velocity V by a factor of about 4, since ${\ensuremath{\lambda}}^{2}$V\ensuremath{\simeq}const---then tilted lamellae should appear as extended states and not as ``solitons.'' We find here that parity-broken states exist for hypereutectic as well as for hypoeutectic and eutectic compositions. We have extended the derivation of the similarity equation derived previously [K. Kassner and C. Misbah, Phys. Rev. Lett. 66, 445 (1991)] to the present situation. This case involves additional subtleties, due to the loss of reflection symmetry about the growth axis. Among other results, we find that the tilt angle \ensuremath{\varphi} should depend on \ensuremath{\sigma}=${\mathit{d}}_{0}$l/${\ensuremath{\lambda}}^{2}$ and \ensuremath{\chi}=l/${\mathit{l}}_{\mathit{T}}$ only, where ${\mathit{d}}_{0}$, l, and ${\mathit{l}}_{\mathit{T}}$ are the capillary, diffusion, and thermal lengths, respectively, and \ensuremath{\lambda} is the wavelength of the pattern. At large enough growth velocities V, \ensuremath{\varphi}\ensuremath{\simeq}\ensuremath{\varphi}(\ensuremath{\sigma}), while at small V the dependence on \ensuremath{\chi} is strong. These predictions can be tested experimentally.

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